A generalization of Hunt's theorem

Abstract

Let E be a locally compact space with countable base and denote by V a kernel on (E, ~) that satisfies the complete maximum principle, ~ the a-field of universally measurable subsets of E. In this article it is shown (Theorem 2.1) that V is the potential kernel of a Hunt semigroup whenever the following conditions are satisfied: (a) if ~p s ~+ (E) then Vcp is a potential (Definition 1.1); and (b) there exists a subspace 9.1 of ~b(E) such that (1)f~9.I~VIfl bounded; (2) Ec(E)c V(9.1)~9.1; and (3) for all 2 > 0 (I + 2 V)(9.I) is dense in 9A. Further, the semigroup is such that (i) Pt(E~(E))c~ for all t > 0 and (ii) it converges strongly to the identity on ~c(E). If V(E~(E))=Eo(E ) it suffices to take 9d=~(E). Then (b) is satisfied and (a) is trivially satisfied. Consequently, Theorem 2.1 implies Hunt's theorem. In [2-] Hansen also obtained a generalization of Hunt's theorem. It is shown that the hypotheses used by Hansen imply that the hypotheses of Theorem 2.1 are verified (see Theorem 3.2). The following notational conventions are used. The subscripts "b", "c" and "0" denote respectively "bounded", "compact support" and "vanishing at infinity". A superscript " + " indicates "non-negative". These subscripts and superscript are applied to subsets of E(E), the set of continuous real-valued functions on E, and to ~, the set of e-measurable functions on E, where ~ is a a-algebra on E.